In a two dimensional flow the components of velocity are given by u = ax; v = by. The stream lines will be_______.

Concept:

Stream function:

The stream function is defined as a function of space and time such that its partial derivative with respect to any direction gives the velocity component at right angles to this direction. It is denoted by ψ.

Calculation:

Given; u = ax, v = by

We know that; 

ψ = byx + f(y) ................. (1)

Differentiate equation (1) w.r.t. x

we know that, , 

Integrate the above equation w.r.t. y

Value of f(y) put in above equation (1)

The streamline equation (expression)  represents a family of curves, specifically hyperbolas in a two-dimensional Cartesian coordinate system. The equation represents a hyperbola when plotted in the xy-plane. 

Hence option (1) is the correct answer.